The image shows a complex number in rectangular form that needs to be converted to polar form. I see three parts:

(a) $\frac{7 - i}{3}$

(b) $-4\pi (4 + i\sqrt{3})$

(c) $(1 + i)^6$

The goal is to find the polar form for each expression.

Let me go through them step by step.

(a) $\frac{7 - i}{3}$

For this, we convert the complex number to polar form by finding its modulus (r) and argument (θ).

Step 1: Compute the modulus $r$

The modulus $r$ is found by using the formula: $r = \sqrt{a^2 + b^2}$ For $7 - i$, $a = 7$ and $b = -1$, so: $r = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50}$ Now, divide by 3 as the number is scaled by 1/3: $r = \frac{\sqrt{50}}{3} = \frac{5\sqrt{2}}{3}$

Step 2: Compute the argument $\theta$

The argument $\theta$ is the angle the complex number makes with the positive real axis. Use the formula: $\theta = \tan^{-1} \left(\frac{b}{a}\right)$ Here, $a = 7$ and $b = -1$, so: $\theta = \tan^{-1} \left(\frac{-1}{7}\right) \approx -0.1419$

So, the polar form of $\frac{7 - i}{3}$ is: $r = \frac{5\sqrt{2}}{3}, \quad \theta \approx -0.1419$

(b) $-4\pi (4 + i\sqrt{3})$

First, simplify the expression inside the parentheses: $-4\pi (4 + i\sqrt{3}) = -16\pi - 4\pi i \sqrt{3}$

Now, find the modulus $r$ and argument $\theta$ for the complex number $-16\pi - 4\pi i\sqrt{3}$.

Step 1: Compute the modulus $r$

$r = \sqrt{(-16\pi)^2 + (-4\pi \sqrt{3})^2} = \sqrt{(256\pi^2) + (48\pi^2)} = \sqrt{304\pi^2} = \pi \sqrt{304}$ Approximating $\sqrt{304}$ gives: $r \approx 54.776$

Step 2: Compute the argument $\theta$

$\theta = \tan^{-1} \left(\frac{-4\pi \sqrt{3}}{-16\pi}\right) = \tan^{-1} \left(\frac{\sqrt{3}}{4}\right) \approx 0.409$

So, the polar form of $-4\pi (4 + i\sqrt{3})$ is: $r \approx 54.776, \quad \theta \approx 0.409$

(c) $(1 + i)^6$

We first convert $1 + i$ to polar form and then raise it to the power of 6.

Step 1: Convert $1 + i$ to polar form

For $1 + i$, the modulus is: $r = \sqrt{1^2 + 1^2} = \sqrt{2}$ The argument is: $\theta = \tan^{-1}(1) = \frac{\pi}{4}$

So, the polar form of $1 + i$ is: $r = \sqrt{2}, \quad \theta = \frac{\pi}{4}$

Step 2: Raise to the power of 6

To raise a complex number in polar form to a power, we use: $r^n \left( \cos(n\theta) + i\sin(n\theta) \right)$ Thus, $(1 + i)^6$ is: $r^6 = (\sqrt{2})^6 = 8, \quad \theta \cdot 6 = \frac{6\pi}{4} = \frac{3\pi}{2}$

So, the polar form of $(1 + i)^6$ is: $r = 8, \quad \theta = \frac{3\pi}{2}$

Would you like further clarification or details on any of these steps?

Here are some related questions for deeper understanding:

1. How do you compute the argument of a complex number in a different quadrant?
2. How does Euler's formula relate to polar form of complex numbers?
3. What is the significance of the argument in trigonometric identities?
4. How do you handle negative complex numbers when finding their polar form?
5. What happens when you raise a complex number to a power in polar form?

Tip: Always ensure to check which quadrant your complex number is in before computing the argument!